LILO is the part of the system that does the booting. When the computer is turned on, it checks several places for boot instructions: floppy, cd-rom and the MBR (not necessarily in that order). When the computer finds one that has bootup instructions, it follows those.
The MBR isn't a program that can tell the computer how to boot, rather it's a place on the hard drive that the computer will look to find instructions. When you put LILO in the MBR, it provides the boot instructions (and as a side-note, it's not the whole LILO executable file that goes into there - it's just a set of instructions that LILO writes to it). If you don't have LILO, or some other bootloader, installed and written to the MBR, the computer won't be able to startup.